84 OUTLINES OF NATURAL PHILOSOPHY. 



these letters denoting the number of teeth in thei* 



respective wheels, therefore 



AxBxC:6xcxd::240:l. 

 Let the number of teeth in the pinions be 7, 5, and 



11, then 



A x B x C : 7 X 5 x 11 : : 240 : 1 ; 



and AxBxC = 7x5xllx 240, or resolving 



240 into its simple factors, 



AxBxC = 7x5x11x3x2x2x2x2x5. 



These factors combined into three different groups 



in any order, will give three products that may be 



substituted for A, B, and C. 

 We may take, for instance, 5x1 1, 7x3x2, and 



2x2x2x5, which will give 55, 42 and 40, for 



the numbers of the teeth that are to work in the 



pinions, where the numbers are 7, 5 and 11, The 



angular velocity of A will thus be to the angular 



velocity of d in the ratio required. 



142. The number of teeth in a wheel, and in a 

 pinion which work together, should be prime to 

 one another, that their coincidences may not recur 

 in the same order after every revolution. 



a. The numbers expressing the ratio of the angular 

 velocity of the wheel to that of the pinion, may 

 sometimes be fractional, and may consist of so ma- 

 ny places, when the fractions are taken away, that, 

 to make the motions exactly in their ratio, would 

 require more wheels, and a greater number of teeth, 

 than can conveniently be allowed. It is then of 



consequence 



